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oblige us to leave it without formal proof. A solid of any elastic substance, isotropic or aeolotropic, bounded by any surfaces presenting projecting edges or angles, or re-entrant angles or edges, however obtuse, cannot experience any finite stress or strain in the neighbourhood of a projecting angle (trihedral, polyhedral, or conical); in the neighbourhood of an edge, can only experience simple longitudinal stress parallel to the neighbouring part of the edge; and generally experiences infinite stress and strain in the neighbourhood of a re-entrant edge or angle; when influenced by any distribution of force, exclusive of surface tractions infinitely near the angles or edges in question. An important application of the last part of this statement is the practical rule, well known in mechanics, that every re-entering edge or angle ought to be rounded to prevent risk of rupture, in solid pieces designed to bear stress. An illustration of these principles is afforded by the complete mathematical solution of the torsion problem for prisins of fan-shaped sections, such as the annexed figures. In the cases corresponding to figures (4), (5), (6) below, the distortion at the centre of the circle vanishes in (4), is finite and determinate in (5), and infinite in (6).

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679. Hence in a rod of isotropic substance the principal axes of flexure ($ 609) coincide with the principal axes of inertia of the area of the normal section; and the corresponding flexural rigidities are the moments of inertia of this area round these axes multiplied by Young's modulus. Analytical investigation leads to the following results, due to St. Venant. Imagine the whole rod divided, parallel to its length, into infinitesimal filaments (prisms when the rod is straight). Each of these contracts or swells laterally with sensibly the same freedom as if it were separated from the rest of the substance, and becomes elongated or shortened in a straight line to the same extent as it is really elongated or shortened in the circular arc which it becomes in the bent rod. The distortion of the cross section by which these changes of lateral dimensions are necessarily accompanied is illustrated in the annexed diagram, in which either the whole normal section of a rectangular b'eam, or a rectangular area in the normal section of a beam of any figure, is represented in its strained and unstrained figures, with the central point o common to the two. The flexure is in planes perpendicular to YO Y,, and concave upwards (or towards X); G the centre of curvature, being in the direction indicated, but too far to be included in the diagram. The straight

sides AC, BD, and all straight lines parallel to them, of the unstrained rectangular area become concentric arcs of circles concave in the

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opposite direction, their centre of curvature, H, being for rods of gelatinous substance, or of glass or metal, from 2 to 4 times as far from O on one side as G is on the other. Thus the originally plane sides AC, BD of a rectangular bar become anticlastic surfaces, of curvatures - and, in the two principal sections. A fat rectangular, or a square, rod of India-rubber [for which o amounts ($ 655) to very nearly ], and which is susceptible of very great amounts of strain without utter loss of corresponding elastic action], exhibits this phenomenon remarkably well.

680. The conditional limitation (5605) of the curvature to being very small in comparison with that of a circle of radius equal to the greatest diameter of the normal section (not obviously necessary, and indeed not generally known to be necessary, we believe, when the greatest diameter is perpendicular to the plane of curvature) now receives its full explanation. For unless the breadth, AC, of the bar (or diameter perpendicular to the plane of flexure) be very small in comparison with the mean proportional between the radius, OH, and the thickness, AB, the distances from Oy to the corners A', C' would fall short of the half thickness, OE, and the distances to B', D' would exceed it by differences comparable with its own amount. This would give rise to sensibly less and greater shortenings and stretchings

in the filaments towards the corners, and so vitiate the solution. Unhappily mathematicians have not hitherto succeeded in solving, possibly not even tried to solve, the beautiful problem thus presented by the flexure of a broad very thin band (such as a watch spring) into a circle of radius comparable with a third proportional to its thickness and its breadth.

681. But, provided the radius of curvature of the lexure is not only a large multiple of the greatest diameter, but also of a third proportional to the diameters in and perpendicular to the plane of fexure; then however great may be the ratio of the greatest diameter to the least, the preceding solution is applicable; and it is remarkable that the necessary distortion of the normal section (illustrated in the diagram of $ 679) does not sensibly impede the free lateral contractions and expansions in the filaments, even in the case of a broad thin lamina (whether of precisely rectangular section, or of unequal thicknesses in different parts).

682. In our sections on hydrostatics, the problem of finding the deformation produced in a spheroid of incompressible liquid by a given disturbing force will be solved; and then we shall consider the application of the preceding methods to an elastic solid sphere in their bearing on the theory of the tides and the rigidity of the earth. This proposed application, however, reminds us of a general remark of great practical importance, with which we shall leave elastic solids for the present. Considering different elastic solids of similar substance and similar shapes, we see that if by forces applied to them in any way they are similarly strained, the surface tractions in or across similarly situated elements of surface, whether of their boundaries of of surfaces imagined as cutting through their substances, must be equal, reckoned as usual per unit of area. Hence; the force across, or in, any such surface, being resolved into components parallel to any directions; the whole amounts of each such component for similar surfaces of the different bodies are in proportion to the squares of their lineal dimensions. Hence, if equilibrated similarly under the action of gravity, or of their kinetic reactions ($ 230) against equal accelerations (8 32), the greater body would be more strained than the less; as the amounts of gravity or of kinetic reaction of similar portions of them are as the cubes of their linear dimensions. Definitively, the strains at similarly situated points of the bodies will be in simple proportion to their linear dimensions, and the displacements will be as the squares of these lines, provided that there is no strain in any part of any of them too great to allow the principle of superposition to hold with sufficient exactness, and that no part is turned through more than a very small angle relatively to any other part. To illustrate by a single example, let us consider a uniform long, thin, round rod held horizontally by its middle. Let its substance be homogeneous, of density p, and Young's modulus, M; and let its length, l, be p times its diameter. Then (as the moment of inertia of a circular area of radius r round a diameter is tar") the

- ()

M flexural rigidity of the rod will (8 679) be

This gives us

4 for the curvature at the middle of the rod the elongation and con. traction' where greatest, that is, at the highest and lowest points of the normal section through the middle point; and the droop of the ends; the following expressions

2pp. plp




and P?l

Thus, for a rod whose length is 200 times its diameter, if its substance be iron or steel, for which p= 7*75, and M = 194 x 10' grammes per square centimetre, the maximum elongation and contraction (being at the top and bottom of the middle section where it is held) are each equal to 8 x 10- xl, and the droop of its ends 2 x 10-6x8. Thus a steel or iron wire, ten centimetres long, and half a millimetre in diameter, held horizontally by its middle, would experience only '000008 of maximum elongation and contraction, and only 002 of a centimetre of droop in its ends: a round steel rod, of half a centi. metre in diameter, and one metre long, would experience o00o8 of máximum elongation and contraction, and 2 of a centimetre of droop: à round steel rod, of ten centimetres diameter, and twenty metres long, must be of remarkable temper (see Properties of Matter) to bear being held by the middle without taking a very sensible permanent set: and it is probable that no temper of steel is high enough in a round shaft forty metres long, if only two decimetres in dia. meter, to allow it to be held by its middle without either bending it to some great angle, and beyond all appearance of elasticity, or breaking it.

683. In passing from the dynamics of perfectly elastic solids to abstract hydrodynamics, or the dynamics of perfect fluids, it is convenient and instructive to anticipate slightly some of the views as to intermediate properties observed in real solids and fluids, which, according to the general plan proposed (8 402) for our work, will be examined with more detail under Properties of Matter.

By induction from a great variety of observed phenomena, we are compelled to conclude that no change of volume or of shape can be produced in any kind of matter without dissipation of energy ($ 247); so that if in any case there is a return to the primitive configuration, some amount (however small) of work is always required to compensate the energy dissipated away, and restore the body to the same physical and the same palpably kinetic condition as that in which it was given. We have seen ($ 643), by anticipating something of thermodynamic principles, how such dissipation is inevitable, even in dealing with the absolutely perfect elasticity of volume presented by every Guid, and possibly by some solids, as, for instance, homogeneous crystals. But in metals, glass, porcelain, natural stones, wood, India. rubber, homogeneous jelly, silk fibre, ivory, etc., a distinct frictional

resistance against every change of shape is, as we shall see later, under Properties of Malter, demonstrated by many experiments, and is found to depend on the speed with which the change of shape is made. A very remarkable and obvious proof of frictional resistance to change of shape in ordinary solids, is afforded by the gradual, more or less rapid, subsidence of vibrations of elastic solids; marvellously rapid in India-rubber, and even in homogeneous jelly; less rapid in glass and metal springs, but still demonstrably, much more rapid than can be accounted for by the resistance of the air. This molecular friction in elastic solids may be properly called viscosity of solids, because, as being an internal resistance to change of shape depending on the rapidity of the change, it must be classed with fluid molecular friction, which by general consent is called viscosity of fuids. But, at the same time, we feel bound to remark that the word viscosity, as used hitherto by the best writers, when solids or heterogeneous semisolid-semifluid masses are referred to, has not been distinctly applied to molecular friction, especially not to the molecular friction of a highly elastic solid within its limits of high elasticity, but has rather been employed to designate a property of slow, continual yielding through very great, or altogether unlimited, extent of change of shape, under the action of continued stress. It is in this sense that Forbes, for instance, has used the word in stating that 'Viscous Theory of Glacial Motion' which he demonstrated by his grand observations on glaciers. As, however, he, and many other writers after him, have used the words plasticity and plastic, both with refer. ence to homogeneous solids (such as wax or pitch, even though also brittle; soft metals; etc.), and to heterogeneous semisolid-semifluid masses (as mud, moist earth, mortar, glacial ice, etc.), to designate the property', common to all those cases, of experiencing, under continued stress either quite continued and unlimited change of shape, or gradually very great change at a diminishing (asymptotic) rate through infinite time; and as the use of the term plasticity. implies no more than does viscosity, any physical theory or explanation of the property, the word viscosity is without inconvenience left available for the definition we have given of it above.

684. A perfect fluid, or (as we shall call it) a fluid, is an unrealizable conception, like a rigid, or a smooth, body: it is defined as a body incapable of resisting a change of shape : and therefore incapable of experiencing distorting or tangential stress (8 640). Hence its pressure on any surface, whether of a solid or of a contiguous portion of

See Proceedings of the Royal Society, May 1865, 'On the Viscosity and Elasticity of Metals' (W. Thomson).

3. Some confusion of ideas might have been avoided on the part of writers who have professedly objected to Forbes' theory while really objecting only (and we believe groundlessly) to his usage of the word viscosity, if they had paused to consider that no one physical explanation can hold for those several cases; and that Forbes' theory is merely the proof by observation that glaciers have the property that mud (heterogeneous), mortar (heterogeneous), pitch (homogeneous), water (homogeneous), all have of changing shape indefinitely and continuously under the action of continued stress.

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